Kempner function

In mathematics, the Smarandache function is a string or a number-theoretic function which is related to the faculty. Historically, it was first used by Édouard Lucas (1883 ), Joseph Neuberg (1887 ) and Aubrey J. Kempner (1918 ) is considered. In 1980 it was "rediscovered" by Florentin Smarandache.

  • 3.1 Pseudo Maran roof function
  • 3.2 Smarandache double factorial function
  • 3.3 Smarandache function with primorial
  • 3.4 Smarandache - Kurepa function and Smarandache - Wagstaff function
  • 3.5 Smarandache Ceil function

Definition and values

The Smarandache function is defined as the smallest natural number for which shares the Faculty of.

Thus, formally is the smallest natural number for which holds

Examples

For example, if the value sought is the smallest of the numbers 1! 2! , ... Looking for 3! That is divisible by 8. Since and and are not divisible by eight, but is.

However, as since the number 7, none of the numbers 1? , 2! ... 6! shares, while 7! trivially divides.

The first values ​​are:

(*) The value is also defined as 0 by some authors.

Properties

Trivially true

Since shares in any case.

A basic result is that equality in the above inequality occurs exactly for prime or:

Proof:

: Be and not prime. Then to show. Since it is not prime, there is with natural numbers. If even so would and would obtain the contradiction. So and so. If so followed, so and so, and you would again the contradiction. Therefore, be and it follows that.

Is prim, it shall not count for as n by def. not in m! occurs. Therefore applies. is clear.

By the way, this results in for the number of primes less than or equal and the integer function:

After Paul Erdős is consistent with the largest prime factor of agreement for asymptotically almost all, ie the number of integers less than or equal, for this does not apply is O (n).

In general, further

And

Where stands for the largest prime factor of.

The general rule

Also applies to (even) perfect numbers ()

Modifications

Pseudo Maran roof function

The pseudo Maran roof function is the smallest integer for which

So the smallest natural to apply to the

(see also triangular number, Gaussian sum formula )

The first values ​​are

Some properties:

  • Are unlimited upward
  • Has infinitely many solutions for
  • Converges for all

Smarandache double factorial function

If we replace in the definition of the factorial by the double factorial

So is

The first values ​​are for

Smarandache function with primorial

The primorial (also Primfakultät, ) is the product of the primes less than or equal to the given number. The Smarandache Near- to- primorial function of is then the smallest prime number is for, or divisible.

Smarandache - Kurepa function and Smarandache - Wagstaff function

For the Smarandache - Kurepa function can not convert the faculty from the double factorial but to the following function:

For prime is analogous to the smallest natural number such that is divisible by.

The first values ​​are 2, 4, 6, 6, 5, 7, 7, 12, 22, 16, 55, and in consequence form A049041 OEIS.

The Smarandache - Wagstaff function used instead

Smarandache Ceil function

The Smarandache - Wagstaff - function k -th order finally is the smallest natural number defined for that is divisible by.

The first values ​​:

More

  • Tutescu suspected that the values ​​of the Smarandache function are always different for two consecutive numbers:
  • There are quite a large variety convergent series using the Smarandache function. Such limits are often called Smarandache constants - not to be confused with the Smarandache constant in the generalized Andricaschen guess.

Credentials

  • Eric W. Weisstein: Smarandache Function. In: MathWorld (English).
  • Kenichiro Kashihara: Comments and topics on Smarandache notions and problems Erhus University Press 1996, ISBN 1-87958-555-3 - PDF
  • Norbert Hungerbühler, Ernst Specker: A generalization of the Smarandache Function to Several Variables. Electronic Journal of Combinatorical Number Theory 6 (2006), # A23 - PDF
  • C. Dumitrescu, N. Virlan, St. Zamfir, E. Radescu, N. Radescu, F.Smarandache: Smarandache Type Function Obtained by Duality, Studii si Cercetari Stiintifice, Seria: Matematica, University of Bacau, no. 9, pp. 49-72, 1999 arXiv: . 0706.2858
  • Sebastian Martin Ruiz, ML Perez: Properties and Problems related to Smarandache Type Functions. In: Mathematics Magazine for grades 1-12. 2/2004 p.46 -53 - arXiv: math/0407479
  • The Smarandache Function Journal, http://fs.gallup.unm.edu/ ~ Smarandache / - Vol 1 (PDF, 1.7 MB), Vol 6 (PDF, 2.6 MB)
  • And Smarandache Notions Journal - Vol 7 (PDF, 5.4 MB), Vol 8 (PDF, 8.8 MB), Vol 9 (PDF, 5.3 MB), Vol 10 (PDF, 7.3 MB), Vol 11 (PDF, 10.8 MB ), Vol 12 (PDF, 12.5 MB), Vol 13 (PDF, 11.1 MB )
  • Number theoretic function
Joseph Jean Baptiste Neuberg Digital Object Identifier Wolfram Research
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